2.4: Material and waveguide dispersion

Modal dispersion (discussed in the previous section) is not the only form of dispersion. In this section, material (chromatic) dispersion will be introduced, however, it should be noted that there are many other forms of dispersion, and all forms of dispersion contribute to the degradation of an optical signal.

Material dispersion is due to the fact that the refractive index of all materials (including air) vary with wavelength. In Figure \(\PageIndex{1}\), the refractive index for fused silica (a highly pure form of glass) and lithium niobate, a material used in optical modulators is shown. It is important to note that the refractive index changes nonlinearly with wavelength.

Figure \(\PageIndex{1}\): Refractive index of fused silica (left) and lithium niobate (right).

For various reasons of usability, refractive index is usually measured, tabulated, and calculated in terms of the wavelength of light, so it makes sense to calculate group velocity and dispersion in terms of wavelength. The group velocity is defined as \(\frac{1}{\frac{\textrm{d}k}{\textrm{d}\omega}}\), by use of the chain rule, the derivative becomes becomes \(\frac{\textrm{d}k}{\textrm{d}\omega} = \frac{\textrm{d}k}{\textrm{d}\lambda}\frac{\textrm{d}\lambda}{\textrm{d}\omega}\). The wavevector is \(k = \frac{2\pi n(\lambda_0)}{\lambda_0}\), where \(\lambda_0\) is the wavelength in free space.

\[\begin{align}\frac{\textrm{d}k}{\textrm{d}\lambda} &= \frac{2\pi}{\lambda_0^2}\left(\lambda_0\frac{\textrm{d}n}{\textrm{d}\lambda} - n(\lambda_0)\right)\\ \lambda_0 &= \frac{2\pi c_0}{\omega}\\ \frac{\textrm{d}\lambda}{\textrm{d}\omega} &= -\frac{2\pi c_0}{\omega^2}\\ \frac{\textrm{d}\lambda}{\textrm{d}\omega} &= -\frac{\lambda^2}{2\pi c_0}\\ \frac{\textrm{d}k}{\textrm{d}\lambda} &= \frac{1}{c_0}\left(n(\lambda_0) - \lambda_0\frac{\textrm{d}n}{\textrm{d}\lambda}\right)\label{eq:invGV}\end{align}\]

Inverting equation \ref{eq:invGV} gives the group velocity as

\[v_g = \frac{c_0}{\left(n(\lambda_0) - \lambda_0\frac{\textrm{d}n}{\textrm{d}\lambda}\right)} \label{eq:GV}\]

As with modal dispersion, changes to the group velocity with frequency (or wavelength) can cause pulse spreading. Consider a material, with a length \(L\) and two pulses with central (angular) frequencies, \(\omega-0.5\delta\omega\) and \(\omega + 0.5\delta\omega\) that propagate through the medium (both with the same starting time). The transit time is \(\tau = L/v_g\). The time difference between the two pulses is

\[\begin{align}\delta t &= \frac{\textrm{d}\tau}{\textrm{d}\omega}\delta\omega\\ &= \frac{\textrm{d}}{\textrm{d}\omega}\left(\frac{1}{v_g}\right)L\delta\omega\\ \delta t &= \left|D_\lambda\right|L\delta\omega\label{eq:matDisp}\end{align}\]

\(D_\lambda\) is the material dispersion coefficient given by

\[D_\lambda = \frac{\textrm{d}}{\textrm{d}\omega}\left(\frac{1}{v_g}\right)\label{eq:freqMatDisp}\]

but this more conveniently expressed in terms of wavelength. By applying the chain rule, we obtain

\[D_\lambda = -\frac{\lambda_0}{c_0}\frac{\textrm{d}^2n}{\textrm{d}\lambda_0^2}\label{eq:ldMatDisp}\]

The dispersion coefficient depends on the second derivative of the change of refractive index with wavelength. This means that materials with no change or a linear change of refractive index with wavelength exhibit no material dispersion. But, quadratic and higher order changes with wavelength will introduce dispersion.

A waveguide, however, consists of a cladding and a core with different refractive indices. This results in an additional contribution called waveguide dispersion, given by

\[D_w = -\left(\frac{1}{2\pi c_0}\right)V^2\frac{\textrm{d}^2k}{\textrm{d}V^2}\label{eq:wgDisp}\]

These two components of dispersion can be used to manipulate the total dispersion to meet system requirements.

Managing dispersion

Figure \(\PageIndex{2}\) shows the group velocity dispersion for a typical single mode fiber. Note that at a wavelength of about 1300 nm, there is no dispersion. However, information requires bandwidth, so there is still a tiny amount of residual dispersion around 1300 nm. However, the wavelength of 1300 nm is not normally used for long-distance links because of nonlinear effects, as discussed in Section \(\PageIndex{3}\). Instead, a longer wavelength (1550 nm) is used, where there is a small amount of positive dispersion (\(D_\lambda \approx\) +20 ps/km.nm. As a result, some form of dispersion compensation is required, which is achieved using optical fibers that have impurities added so that the dispersion coefficient is negative. Typically, a dispersion compensating fiber (DCF) has a very strong dispersion \(D_\lambda \approx\) - 160 ps/km.nm because these fibers are more expensive and have higher losses.

Figure \(\PageIndex{2}\): Group velocity dispersion of a single mode fiber. (Copyright; author via source)

Example \(\PageIndex{1}\)

You are designing an optical link between New York and Amsterdam with a single channel capacity of 40 Gb/s, operating at a wavelength of 1550 nm. Ignoring any electronic repeaters, what is the accumulated pulse spreading? Is the pulse spreading within the specification for communications? About how much (DCF) will be required to compensate for the pulse spreading?

Solution

• distance, \(L\) from Amsterdam to New York is 5790 km
• required bandwidth is \(\Delta f\) ~80 GHz (twice the data rate), which in wavelength is \(\Delta\lambda = \frac{\lambda^2}{c_0}\Delta f\ =) 0.64 nm

Putting these numbers in leads to a pulse spreading of \(\delta t = D_\lambda\Delta\lambda L\) = 74 ns, while the time-per-bit is 25 ps (even if modern modulation techniques are used, the bit time will not be increased by 1000 times). So, dispersion compensation is required. The length of the fiber is \(\frac{\delta t}{D_\lambda\Delta\lambda} =\) 80 m.

Nonlinear effects

Most optical systems are linear. A simple way to think of the difference between linear and nonlinear is that, in a linear system, the light direction is modified by the medium, but the medium is not modified by the light. In a nonlinear system, the light modifies the medium, which then changes the frequency (wavelength) and intensity of the light. Linearity holds for almost all situations, however, when the light intensity is high (high means kW/cm² or higher), then nonlinear effects become important. A typical single mode optical fiber has a mode field diameter (the area in which the vast majority of the light power is concentrated) of 12 µm, corresponding to an area of about \(1.1\times 10^{-6}\) cm², so a 1 W laser already provides an intensity of 1 MW/cm². More realistically, optical communications systems work with powers in the 10-100 mW range, which leads to intensities of 1-100 kW/cm², chosen to minimize nonlinear effects.

The most immediately relevant nonlinear effect is called four-wave mixing. Four wave mixing creates new frequencies of light based on sums and differences of input light signals. To give an example: to increase the capacity of an optical communications system, it is standard to divide the data into channels that are separated by wavelength/frequency (the separation between channels is about 100 GHz). In a linear system, this is fine, because the two wavelengths will never mix with each other and can be separated using standard techniques that are presented later. So we have two signals at \(\omega_1\) and \(\omega_2 = \omega_1+\delta\omega\). These can mix to create two more signals at \(2\omega_1 -\omega_2 = \omega_1-\delta\omega\) and \(2\omega_2 -\omega_1 = \omega_1+2\delta\omega\). These two signal fall in adjacent channels that might also be in use. The new signals are a mixture of the information carried in channels one and two, so data from (potentially) three channels have been mixed uncontrollably in one channel.

It gets worse though. Now we have four signals that can all mix with each other, and this process can repeat endlessly, eventually corrupting all available channels (see Figure \(\PageIndex{3}\)). What prevents this cascade? Dispersion prevents the cascade. The new signals generated in a four-wave mixing process (and many other nonlinear processes) grow in amplitude as \(E_{new} = E_1^2E_2e^{i\Delta kz}\), where \(E_i\) is the electric field amplitude \(z\) is the distance traveled and the phase-mismatch, \(\Delta k = 2k_1 - k_2\), is the difference in wavevectors for the input waves. If the refractive index varies nonlinearly with wavelength \(|\Delta k|>0\) otherwise \(\Delta k = 0\). If the phase-mismatch is zero, then the new signal will grow in amplitude until all signals are of about equal amplitude. Otherwise, the growth of the new signal is prevented by the oscillation of the complex exponential term. Even with dispersion slowing the growth of nonlinear effects, the intensity of the light still has to be low enough that processes like four-wave mixing are tiny over a half oscillation of the exponential term.

Figure \(\PageIndex{3}\): An illustration of how four-wave mixing can corrupt a data transmission. In the bottom picture, two channels are present, which then mix and add signal to two adjacent channels (which may or may not already have a signal). This process then repeats, spreading the signal across multiple channels (moving from bottom to top in the figure).